Hot Spots: From Smooth to Nonsmooth Simply Connected Planar Domains
Assuming the absence of interior local extrema for first positive Neumann eigenfunctions on bounded smooth simply connected planar domains, as reported in \cite{SmoothInput}, we extend the hot spots conclusion to simply connected domains of finite area with compact Sobolev embedding and simple first positive Neumann eigenvalue. The supremum and infimum are approached in arbitrarily thin boundary layers. For bounded domains and eigenfunctions continuous up to the boundary, both extrema are attained on the boundary. We also transfer polynomial Neumann heat-kernel bounds uniformly along a conformal exhaustion and construct explicit bounded and unbounded examples, including domains with cusps and a domain with infinitely many corners and bottlenecks. Some of these examples admit no polynomial Neumann heat-kernel bound.